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The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).  

The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).  

写成广泛性质（质量，体积，熵……）的函数时，第二定律被证明等价于弱凸函数内能 U。

'''<font color="#32CD32">写成广泛性质（质量，体积，熵……）的函数时，第二定律被证明等价于弱凸函数内能 U。The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).</font>'''

 

==Corollaries 推论==

==Corollaries 推论==

The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.

The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.

The ergodic hypothesis is also important for the Boltzmann approach. <font color = 'green'>It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time.</font> Equivalently, it says that time average and average over the statistical ensemble are the same.

The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.

'''<font color="#ff8000">遍历假设ergodic hypothesis</font>'''对'''<font color="#ff8000">玻尔兹曼方法Boltzmann approach</font>'''也很重要。<font color = 'blue'>遍历假设认为</font>在很长一段时间内，在具有相同能量的微观态相空间的某些区域所花费的时间与这个区域的体积成正比，即在很长一段时间内，所有可访问的微观状态<font color = 'blue'>出现/成立</font>的可能性都是一样的。<font color = 'red'><s>同样的，</s></font><font color = 'blue'>等价于说，</font>它表明时间平均值和统计集合的平均值是相同的。

'''<font color="#ff8000">遍历假设ergodic hypothesis</font>'''对'''<font color="#ff8000">玻尔兹曼方法Boltzmann approach</font>'''也很重要。<font color = 'blue'>遍历假设认为</font>在很长一段时间内，在具有相同能量的微观态相空间的某些区域所花费的时间与这个区域的体积成正比，即在很长一段时间内，所有可访问的微观状态<font color = 'blue'>出现/成立</font>的可能性都是一样的。<font color = 'red'><s>同样的，</s></font><font color = 'blue'>等价于说，</font>它表明时间平均值和统计集合的平均值是相同的。

 

There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref>

There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://entropysite.oxy.edu Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref>

The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.

The [[Poincaré recurrence theorem]] considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.

The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. <font color = 'green'>It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics.</font> Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.

The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics.Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.

庞加莱始态复现定理<font color = 'red'><s>（又叫递归定理）</s></font>考虑了孤立物理系统的理论微观描述。

庞加莱始态复现定理<font color = 'red'><s>（又叫递归定理）</s></font>考虑了孤立物理系统的理论微观描述。

   --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）又叫递归定理吗？存疑

   --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）又叫递归定理吗？存疑

经过热力学作用去除隔离内壁之后，便可以认为其是热力学系统的模型。在足够长的时间后，系统将返回到非常接近初始状态的微观定义状态。庞加莱复现时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且<font color = 'red'><s>敏感地依赖于被热力学作用拆除的墙体的几何形状</s></font><font color = 'blue'>对那个被热力学作用拆除的内壁的几何形状敏感</font>。复现定理可能被认为是明显与热力学第二定律相矛盾的。但是，更显而易见的是，它只是通过移除两个系统之间的内壁而形成的隔离系统中热力学平衡的微观模型。<font color = 'red'><s>对于典型的热力学系统而言，重复时间是如此之长（比宇宙的寿命长很多倍），以至于在所有的实际目的中，人们都无法观察到这种重复。</s></font><font color = 'blue'>对所有典型的热力学系统的实际目的，人们无法观察到如此之长的复现时间很长（比宇宙的寿命长很多倍）。</font>尽管如此，还是有人会想像一个机会可以等待庞加莱复现的出现，然后重新插入被热力学作用去除的内壁。然后很明显，不可逆性的出现是由于<font color = 'red'><s>庞加莱递归的完全不可预测性</s></font><font color = 'blue'>庞加莱复现完全不可预测的特性</font>，因为仅仅给出了初始状态<font color = 'red'><s>遵守热力学平衡的条件之一</s></font><font color = 'blue'>是热力学平衡之一</font>，就像宏观热力学的情况一样。

经过热力学作用去除隔离内壁之后，便可以认为其是热力学系统的模型。在足够长的时间后，系统将返回到非常接近初始状态的微观定义状态。庞加莱复现时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且<font color = 'red'><s>敏感地依赖于被热力学作用拆除的墙体的几何形状</s></font><font color = 'blue'>对那个被热力学作用拆除的内壁的几何形状敏感</font>。复现定理可能被认为是明显与热力学第二定律相矛盾的。但是，更显而易见的是，它只是通过移除两个系统之间的内壁而形成的隔离系统中热力学平衡的微观模型。<font color = 'red'><s>对于典型的热力学系统而言，重复时间是如此之长（比宇宙的寿命长很多倍），以至于在所有的实际目的中，人们都无法观察到这种重复。</s></font><font color = 'blue'>对所有典型的热力学系统的实际目的，人们无法观察到如此之长的复现时间很长（比宇宙的寿命长很多倍）。</font>尽管如此，还是有人会想像一个机会可以等待庞加莱复现的出现，然后重新插入被热力学作用去除的内壁。

然后很明显，不可逆性的出现是由于<font color = 'red'><s>庞加莱递归的完全不可预测性</s></font><font color = 'blue'>庞加莱复现完全不可预测的特性</font>，因为仅仅给出了初始状态<font color = 'red'><s>遵守热力学平衡的条件之一</s></font><font color = 'blue'>是热力学平衡之一</font>，就像宏观热力学的情况一样。

 

 

即使<font color = 'red'><s>可以等待，也没有实际可操作性</s></font><font color = 'blue'>一个人可以等那么长时间，他也没有实际操作的可能性</font>来选择合适的时间重新插入内壁。庞加莱始态复现定理为洛施密特悖论提供了一个解决方案。如果一个孤立的热力学系统能以多倍于平均庞加莱复现时间的长度下<font color = 'red'><s>进行</s></font><font color = 'blue'>被</font>监控，则该系统的热力学行为在时间反转下将变得恒定。

即使<font color = 'red'><s>可以等待，也没有实际可操作性</s></font><font color = 'blue'>一个人可以等那么长时间，他也没有实际操作的可能性</font>来选择合适的时间重新插入内壁。庞加莱始态复现定理为洛施密特悖论提供了一个解决方案。如果一个孤立的热力学系统能以多倍于平均庞加莱复现时间的长度下<font color = 'red'><s>进行</s></font><font color = 'blue'>被</font>监控，则该系统的热力学行为在时间反转下将变得恒定。